Invisibility to EM fields

Recently a device invisible to a static magnetic field was described. The device features a cylinder with two concentric layers. While the inner layer consists of a superconducting material that repels magnetic fields, the outer layer is a ferromagnetic material that attracts them. Placed in a magnetic field, the device has no effect on the field lines, showing neither a shadow nor a reflection. So an object inside the device cannot be detected.www.abc.net.au/science/articles/2012/03/23/3461975.htm

My question is: how can one make it work for light (EM field)? What are the physics /equations involved?

Some materials called "Metamaterials" have been engineered in the last few years that are invisible to microwaves. Use a specific gradient /material/structure pattern to create a negative refractive index. The materials are designed with very precise gradient pattern that is "invisible" to a small portion of microwaves. The problem with doing this with visible light is that we would need to engineer some kind of pattern that could have a negative refractive index for all wavelengths of visible light at the same time, which would prove quite a challenge!

Conductivity is related to refractive index (its complex form), but refractive index is the magnitude prefered in optics. In order to make an object invisible you do need to grade its refractive index, and that's how metamaterials are made.